Projector, Neural, and Tensor-Network Representations of $\mathbb{Z}_N$ Cluster and Dipolar-cluster SPT States

The $\mathbb{Z}_N$ cluster-state wavefunction, a paradigmatic example of symmetry-protected topological (SPT) order with $\mathbb{Z}_N \times \mathbb{Z}_N$ symmetry, is expressed in various equivalent ways. We identify the projector-based scheme called the $P$-representation as the efficient way to express cluster and dipolar cluster state's wavefunctions. Employing the restricted Boltzmann machine scheme to re-write the interaction matrix in the $P$-representation in terms of neural weight matrices allows us to develop the neural quantum state (NQS) and the matrix product state (MPS) representations of the same state. The NQS and MPS representations differ only in the way the weight matrices are split and grouped together in a matrix product. For both $\mathbb{Z}_N$ cluster and dipolar cluster states, we derive in closed form the weight function $W(s,h)$ that couples physical spins $s$ to hidden variables $h$, generalizing the previous construction for $Z_2$ cluster states to $\mathbb{Z}_N$. For the dipolar cluster state protected by two charge and two dipole symmetries, the procedure we have developed leads to the tensor product state (TPS) representation of the wavefunction where each local tensor carries three virtual indices connecting a given site to two nearest neighbors and one further neighbor. We benchmark the resulting TPS construction against conventional MPS representation using density-matrix renormalization group simulations and argue that the TPS could offer a more efficient representation for some modulated SPT states. As a by-product of the investigation, we generalize the previous $Z_2$ matrix product operator construction of the Kramers-Wannier (KW) operator to $\mathbb{Z}_N$ and interprets it as the dipolar generalization of the discrete Fourier transform on $\mathbb{Z}_N$ variables. The new interpretation naturally explains why the KW map is non-invertible.